Mass threshold of the limit behavior of normalized solutions to Schrödinger equations with combined nonlinearities

This paper aims to give an affirmative answer to a conjecture raised by Soave (2020) [25] and considers the qualitative properties of normalized solutions to Sobolev critical/subcritical Schrödinger equations with combined nonlinearities. Precisely, we establish the mass threshold a¯ such that the mountain pass type normalized solution exists for the Sobolev critical/subcritical Schrödinger equation with combined mass critical and mass supercritical nonlinearities. We then show that a¯ is also a threshold of the limit behavior of the mountain pass type normalized solution of the Schrödinger equation with combined nonlinearities as the exponent of lower order term tending to the mass critical exponent. Among which, the results in the case that the mass small than the threshold a¯ give an affirmative answer to the conjecture raised by Soave.